Combinatorial algorithms are widely used for decision-making and knowledge discovery, and it is important to ensure that their output remains stable even when subjected to small perturbations in the input. Failure to do so can lead to several problems, including costly decisions, reduced user trust, potential security concerns, and lack of replicability. Unfortunately, many fundamental combinatorial algorithms are vulnerable to small input perturbations. To address the impact of input perturbations on algorithms for weighted graph problems, Kumabe and Yoshida (FOCS'23) recently introduced the concept of Lipschitz continuity of algorithms. This work explores this approach and designs Lipschitz continuous algorithms for covering problems, such as the minimum vertex cover, set cover, and feedback vertex set problems. Our algorithm for the feedback vertex set problem is based on linear programming, and in the rounding process, we develop and use a technique called cycle sparsification, which may be of independent interest.

翻译：组合算法被广泛用于决策制定与知识发现，确保其输出即使在输入受到微小扰动时也能保持稳定至关重要。若无法做到这一点，可能导致诸多问题，包括代价高昂的决策、用户信任度降低、潜在的安全隐患以及可复现性缺失。遗憾的是，许多基础组合算法在面对微小的输入扰动时表现脆弱。为应对输入扰动对加权图问题算法的影响，Kumabe与Yoshida（FOCS'23）近期引入了算法Lipschitz连续性的概念。本研究沿此思路展开，为覆盖问题——如最小顶点覆盖、集合覆盖及反馈顶点集问题——设计了Lipschitz连续算法。我们针对反馈顶点集问题的算法基于线性规划，并在舍入过程中开发并应用了一种称为环稀疏化的技术，该技术本身可能具有独立的研究价值。